> Exponents and Roots
Exponents and Roots

Exponents are used to denote the repeated the multiplication of a number by itself; for e.g. = (3)(3)(3)(3) = 81 . In the expression 3^{4}, 3 is called the base, 4 is called the exponent, and we read the expression as "3 to the fourth power." When the exponent is 2, we call the process squaring. Thus 7 squared is 49, 7^{2 }= (7)(7) = 49.

A negative number raised to an even power is always positive, and a negative number raised to an odd power is always negative. Note that without the parentheses, the expression 3^{2} means "the negative of '3 squared'"; that is, the exponent is applied before the negative sign. So (3)^{2} = 9, but 3^{2} = 9.

Exponents can also be negative or zero; such exponents are defined as follows.

For all nonzero numbers a, a^{0} = 1. The expression 0^{0}is undefined.

For all nonzero numbers a, , etc. Note that .

A square root of a nonnegative number is a number such that . For e.g. 4 is a square root of 16 because . Another square root of 16 is 4, since . All positive numbers have two square roots, one positive and one negative. The only square root of 0 is 0. The symbol is used to denote the nonnegative square root of the nonnegative number . Therefore, and . Square roots of negative numbers are not defined in the real number system.

Here are some important rules regarding operations with square roots, where and

A square root is a root of order 2. Higher order roots of a positive number are defined similarly. For orders 3 and 4, the cube root and fourth root represent numbers such that when they are raised to the powers 3 and 4 respectively, the result is These roots obey rules similar to those above (but with the exponent 2 replaced by 3 or 4 in the first two rules).

For oddorder roots, there is exactly one root for every number , even when is negative.

For evenorder roots, there are exactly two roots for every positive number and no roots for any negative number

For e.g. 8 has exactly one cube root, but 8 has two fourth roots and ; and 8 has exactly one cube root, , but 8 has no fourth root, since it is negative.

